Problem: What is the slope of the line tangent to $f(x) = 2x^{2}+2x-3$ at $x = 0$ ?
The slope of the tangent line is $ \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(2(x+\Delta x)^{2}+2(x+\Delta x)-3) - (2x^{2}+2x-3)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(2(x^{2}+2x \Delta x+\Delta x^{2})+2(x+\Delta x)-3) - (2x^{2}+2x-3)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{2x^{2}+4(x \Delta x)+2\Delta x^{2}+2x+2(\Delta x)-3-2x^{2}-2x+3}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{4(x \Delta x)+2\Delta x^{2}+2(\Delta x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} 4x+2(\Delta x)+2$ $ = 4x+2$ $ = (4)(0)+2$ $ = 2$